Wu Huishan: Avoiding degrees of orders on torsion-free abelian groups

Invitation to the Logic Seminar at the National University of Singapore

Date: Wednesday, 2 November 2016, 17:00 hrs

Room: S17#04-05, Department of Mathematics, NUS

Speaker: Wu Huishan, Nanyang Technological University

Title: Avoiding degrees of orders on torsion-free abelian groups.

URL: http://www.comp.nus.edu.sg/~fstephan/logicseminar.html

An abelian group admits an order iff it is torsion-free,
we consider degrees of orders on computable torsion-free abelian groups.
For a computable torsionfree abelian group G, Solomon (2003) showed
that if G has rank 1, then it has exactly two computable orders;
if G has finite rank 2 or more, then it has orders of all Turing
degrees; and if G has infinite rank, then it has orders of
all degrees greater equal 0′.
Motivated by the question whether the set of all degrees of orders
on a computable torsion-free abelian group is closed upwards,
Kach, Lange and Solomon (2013) constructed a computable
torsion-free abelian group G of infinite rank with exactly
two computable orders and a noncomputable, computably enumerable
set C such that every C-computable order on G
is computable, so this G has no orders in deg(C) > 0,
but has orders in 0, a negative answer is provided for
above question.
One proposed research topic is to study the computational complexity
of above computably enumerable set C from the standpoint of
classical computability theory, we will present a recent work on
this topic from the viewpoint of high/low hierarchy.
This is a joint work with my supervisor Wu Guohua.

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